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            算法03：回归算法
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            <p><strong>回归算法</strong>（Regression）是研究自变量和因变量之间关系的一种预测模型技术。回归算法与分类算法类似，是基于已有的数据样本，对未来的样本进行预测。回归算法与分类算法也有不同，主要区别在于输出变量的类型。回归是定量的输出，即连续变量的预测；分类是定性的输出，即离散变量的预测。例如：预测明天的温度，是回归任务；预测明天是晴天还是雨天，是分类任务。回归算法可以通过三种方法分类：自变量的个数、因变量的类型和回归线的形状，主要包括线性回归、逻辑回归、多项式回归、逐步回归等多种算法。</p>
<a id="more"></a>
<p><strong>均值（mean）</strong>：又称平均数或平均值</p>
<p>$$x=\cfrac{\sum_{i=1}^n x_i}{n}$$</p>
<p><strong>中位数（median）</strong>：将数据中各数值按照大小顺序排列，位于中间位置的变量</p>
<p><strong>众数（mode）</strong>：数据中出现次数最多的数</p>
<p><strong>方差（variance）</strong>：描述离散程度的衡量方式，此处的$x$为平均数</p>
<p>$$s^2=\cfrac{\sum_{i=1}^n (x_i-x)^2}{n}$$</p>
<p><strong>标准差（standard deviation）</strong>：将方差开方就得到标准差</p>
<hr>
<h1 id="线性回归（Linear-Regression）"><a href="#线性回归（Linear-Regression）" class="headerlink" title="线性回归（Linear Regression）"></a>线性回归（Linear Regression）</h1><p>对于一组分布在二维坐标系上的样本，线性回归的目标是找到一条描述样本变化规律的线，使所有样本尽可能接近这条线。</p>
<p><img src="lr-01.png" alt=""></p>
<p>例如：找到房屋的面积、房间数对价格的影响，现有以下数据：</p>
<table>
<thead>
<tr>
<th style="text-align:center">面积</th>
<th style="text-align:center">房间数</th>
<th style="text-align:center">价格</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">210</td>
<td style="text-align:center">3</td>
<td style="text-align:center">40.0</td>
</tr>
<tr>
<td style="text-align:center">160</td>
<td style="text-align:center">3</td>
<td style="text-align:center">33.1</td>
</tr>
<tr>
<td style="text-align:center">240</td>
<td style="text-align:center">3</td>
<td style="text-align:center">36.9</td>
</tr>
<tr>
<td style="text-align:center">141</td>
<td style="text-align:center">2</td>
<td style="text-align:center">23.2</td>
</tr>
<tr>
<td style="text-align:center">300</td>
<td style="text-align:center">4</td>
<td style="text-align:center">54.0</td>
</tr>
</tbody>
</table>
<p>假设有一个公式，能够根据面积和房间数算出价格，这个公式不仅满足现在的样本数据，在新的数据上也要尽量准确，假定这个公式是线性函数：</p>
<p>$$h(x)=\theta_0+\theta_1 x_1+\theta_2 x_2$$</p>
<p>其中，$x_1$和$x_2$分别表示面积和房间数，$\theta$表示权重，$h(x)$是房屋的价格。</p>
<p>首先把他看作一条$y=wx+b$的直线，其中$w$就是权重$\theta$，$x$为样本数据，$b$为截距，为了公式推导的方便，我们设$x_0=1$，上面的公式简化如下：</p>
<p>$$h(x)=\theta_0 x_0+\theta_1 x_1+\theta_2 x_2=\sum_{i=0}^n \theta_i x_i=\theta^Tx$$</p>
<p>现在只需得到$\theta$，即可得到最终的模型。想要模型能够准确描述当前数据，最理想的情况是这条线能通过所有样本数据，但实际上这几乎不可能，所以只能使$\theta$能够在当前数据集上尽可能准确。反过来想，当找到这条直线后，每个样本距离这条直线都存在一定的误差，即截距，当总截距最小时，这条直线就是我们需要的模型，所以我们只要找到截距与模型间的关系，就可以获得模型，这个关系被称为<strong>损失函数</strong>。</p>
<p>$$J(\theta)=\cfrac{1}{2m}\sum_{i=1}^m (h(x^{(i)})-y^{(i)})^2$$</p>
<p>其中，$m$为样本数量，按照这个公式，求解出使损失函数最小的$\theta$。</p>
<hr>
<h1 id="逻辑回归（Logistic-Regression）"><a href="#逻辑回归（Logistic-Regression）" class="headerlink" title="逻辑回归（Logistic Regression）"></a>逻辑回归（Logistic Regression）</h1><p>逻辑回归就是将上面线性回归预测的值，转化为离散的结果进行输出。逻辑回归虽然名字里带回归，实际上是分类，只是过程中用到了回归算法。逻辑回归的具体算法此处暂不详述。</p>

        
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